Assets, Liabilities And Risk

29

ORiON, Vol. 17, No. 1/2, pp. 29-54 ISSN 0259-191-X

ASSETS, LIABILITIES AND RISK

ROB THOMSON

Department of Statistics and Actuarial Science

University of the Witwatersrand

Johannesburg

South Africa

ABSTRACT

Financial economists and actuaries do not always talk the same language. One particular

difference of concern to actuaries is the method of treatment (or non-treatment) of the

liabilities of an investor in the portfolio selection problem. Another difference relates to the

way in which liabilities are valued. In this paper, these differences are discussed and possible

ways forward are suggested.

1. INTRODUCTION

In the development of portfolio theory, Markowitz (1952, 1959) made no reference to the

liabilities of the investor. The reasons were presumably twofold: in the first place, it could be

assumed that a liability was merely a short position in one of the assets in the opportunity set,

and secondly it could be assumed that that asset, like all the others in that set, was tradable.

The Capital Asset Pricing Model (CAPM) (Tobin, 1958; Sharpe, 1964; Lintner, 1965;

Mossin, 1966) was also developed without reference to the liabilities of investors, mainly

because its authors were concerned with capital assets (i.e. assets in non-zero supply in the

economy) or at least shares and other financial instruments providing entitlement to the

proceeds of such assets. The use of the CAPM to define a ‘market price of risk’ resulted in a

concept of risk without reference to the investor’s liabilities. Subsequent measures of risk such

as Sharpe’s (1966) measure, Treynor’s (1966) measure, Jensen‘s (1968, 1969) measure and the

appraisal ratio (Bodie, Kane & Marcus, 1996: 780) follow suit.

Borch (1990) seeks to bridge the gap between financial economists and actuaries with regard

to problems related to insurance. But his interest in the portfolio selection problem relates

more to insurance and reinsurance portfolios than to capital asset portfolios. And his interest

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in the pricing of liabilities relates more to the development of explanatory models of market

prices than to the determination of market-based values for complex portfolios of liabilities

that are not explicitly priced in the market.

This paper discusses the importance of the liabilities of an investor in the discussion of risk,

as well as the quantification of risk for the purposes of the valuation of the liabilities. Section

2 considers the former matter from the perspective of financial economics and section 3 from

the point of view of fund management. Section 4 discusses the use of benchmark portfolios

based on asset and liability modelling. Section 5 compares this approach with that of the

CAPM. In section 6, consideration is given to the determination of the value of the liabilities.

In order to justify reference to the liabilities of the financial institution, as opposed to the

preferences of prospective beneficiaries, in the asset allocation problem, the principal-agent

problem is discussed in section 7, as well as the trusteeship function. Section 8 concludes.

2. THE PLACE OF LIABILITIES IN FINANCIAL ECONOMICS

Both portfolio theory and the CAPM are based on the expected utility theory of Von

Neumann & Morgenstern (1947). Expected utility theory, in turn, is based on the distribution

of the outcomes and the utility function of the decision-maker. Now suppose, for simplicity,

that the decision-maker is an investor with assets and liabilities maturing at a specified time

horizon. It must be assumed that the decision-maker will be indifferent between an extra rand

of asset proceeds and one rand less of liability payments at the time horizon. If the future

proceeds of assets available to the decision-maker are correlated with the future payments for

which it will become liable, it is the net future proceeds whose distribution must be

considered. All else being equal, if the investor invests in assets that are positively correlated

with its liabilities, the risks of low net proceeds at the time horizon will be reduced and vice

versa. Conversely, if the investor invests part of its wealth in assets that are positively

correlated with its other assets, the risks of low proceeds at the time horizon are increased and

vice versa. And indeed, these results follow from portfolio theory.

Suppose, for example, that an investor has fixed exposure k (which may be negative) to

security 1, and

α

l and (1 –

α

)l to securities 2 and 3 respectively, where k and l (> 0) are

specified constants and

α

is the decision variable. Suppose that the values of the securities at

the time horizon are jointly distributed with mean

31













=

3

2

1

µ

and covariance matrix













=Σ

333231

232221

131211

σσσ

.

Suppose also that the investor has a quadratic utility function.

Then it may be shown that the value of

α

that maximises the investor’s expected utility is

)( 12131

σ

µ

α

−

+

=

ckba

where a, b and c are functions of 33232232 ,,,,

σ

µ

, and c > 0.

Thus, if k is positive (so that security 1 is an asset), then the optimal exposure to security 2

increases as the covariance of the values of securities 1 and 3 increases relative to that of

securities 1 and 2, and vice versa. If k is negative (so that security 1 is a liability) then the

converse holds.

The assumption of quadratic utility is used to simplify the above analysis. Utility functions of

higher order would introduce higher moments than covariances into the relationships

involved. But the above analysis is sufficient to show that, in general, it is the expected utility

of the entire net wealth of the decision-maker that must be maximised, not the expected utility

of that portion of the decision-maker’s gross wealth which has been allocated to a particular

investment manager.

Investment practitioners frequently abuse the theory by ignoring investors’ liabilities.

Typically, linked internal rates of return on investments are determined for investment

performance measurement and compared either with the return on an appropriate index or

with the returns achieved by other investment managers. (Adams, 1989: 310-6; Bodie, Kane

& Marcus, 1996: 778) These comparisons are invariably made without reference to the

investors’ liabilities. Even where risk-adjusted investment returns are compared, liabilities are

typically ignored. (Bodie, Kane & Marcus, 1996: 779-88) Academics and other researchers

have followed suit (e.g., in South Africa, Patrick & Ward, 1996 and Swart, 1999).

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Adams (1989: 9) distinguishes between financial economists’ and actuaries’ definitions of

risk. Whereas a financial economist, he suggests, might define risk to be ‘uncertainty of future

returns’ on investments, an actuary would define it with reference to the investor’s ‘ability to

meet liabilities when they fall due’. This distinction may be true in practice, but if so it is not

portfolio theory that is at fault. Thomson (1998) defended the use of expected utility theory

for normative purposes in the context of defined-contribution retirement funds. Although

mean-variance analysis has its faults—and the necessary caveats are clearly stated and

discussed in Markowitz (1959)—it is still useful as a first approximation to the portfolio

selection problem, provided the liabilities of the investor are taken into account. The

necessary caveats were similarly stated by Sharpe (1964) with regard to the CAPM, but they

have been largely ignored by practitioners, despite numerous findings that the underlying

assumptions do not hold in practice. This matter is discussed further in section 5 below.

3. THE PLACE OF LIABILITIES IN FUND MANAGEMENT

In investment management parlance, ‘fund management’ refers only to the assets of a fund. In

practice the liabilities of a fund may be beyond the control even of the trustees, let alone the

investment managers. But this does not mean that the liabilities can be ignored in the

management of the assets.

Leibowitz (1987) introduces the concept of a ‘return on liabilities’, which can be calculated

retrospectively in much the same way as a return on assets. Whilst this concept has some

explanatory value, it does not mean anything on its own. It can be compared with the return

on assets, but without knowledge of the relative values of the assets and liabilities the net

effect is unclear. What is needed, therefore, is to focus on the surplus (i.e. the value of the

assets less the value of the liabilities) in monetary terms—in other words, on the net wealth of

the fund.

If an investor’s net wealth W(t) is positive at time t = 0 then its value at time t > 0 is a positive

linear function of the return I(t) earned up to that time; i.e.

)}(1){0()( tIWtW

+

=

.

Under these circumstances it would therefore be consistent with portfolio theory and the

CAPM to measure investment performance in terms of the returns earned on net wealth.

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However, contrary to the assumptions of portfolio theory and the CAPM, liabilities cannot be

regarded as short positions in tradable assets. Otherwise, investment managers could be given

control over the liabilities of investors as well as their assets, and financial institutions would

be unnecessary. In the first place, if a liability is transferred from the original counterparty to

another, the counterparty risks associated with the liability are changed, and the value of the

liability changes accordingly. For example, if a retirement fund transfers its liability for a

final-salary pension to a third party, the third party may be unable to meet the liability when it

becomes payable.

On the other hand, the moral hazard associated with the liability may increase enormously on

such transfer, thus increasing its value. This is particularly true where the moral hazard

associated with a particular counterparty is controlled by a convergence of interest between

the institution and that counterparty. For example, if a retirement fund transfers its liability for

a final-salary pension to a third party, the employer may substantially increase the employee’s

salary at the end of the employee’s career, thus increasing the value of the liability without

any cost to the employer. In this case the institution is the retirement fund and the

counterparty is the employer. The increase in the value of the liability arises because the

counterparty no longer has an interest in minimising the cost of funding the benefits.

In terms of the Pension Funds Act (Act no. 24 of 1956 as amended), a retirement fund

member may not cede or assign his or her rights to benefits from a fund, and a transfer of

liabilities from a retirement fund is subject to strict regulation.

Retirement funds may be able to insure some of their benefits— specifically those for which

there is little or no moral hazard— but this leaves the core liabilities of a final-salary

retirement fund, namely the normal retirement benefits, uninsurable and untradable.

The liabilities of life offices are less circumscribed. Life offices are able to reinsure their

liabilities to a large extent. This means that, within the reinsurance market, they are tradable.

However, the reinsurance market is generally distinct from the securities markets. Although

reinsurers are increasingly providing products that modify the investment risks of insurers,

different managers are involved in the management of the assets of the life office and must be

separately mandated and measured.

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In retirement funds, thanks to the popularity of split funding, not only does an investment

manager not have control over the liabilities, it may not even have control over all the assets.

How, then, do we take cognisance of the entire net wealth of the fund in managing its risks?

4. BENCHMARK PORTFOLIOS BASED ON ASSET AND LIABILITY

MODELLING

In the light of the above considerations, it is therefore becoming increasingly widely

advocated that investment managers should be measured against a benchmark portfolio

constructed to reflect the liabilities of the investor. A revised regulation 28 has been drafted

by the Financial Services Board to allow for this process in terms of the Pension Funds Act.

And the Actuarial Society of South Africa is currently working on a new guidance note on

this subject for actuaries advising the trustees of retirement funds. One of the research

projects of the Actuarial Society relates to the capital adequacy requirements of South African

life offices. This project will necessitate the stochastic modelling of the assets and liabilities

so as to develop measures of capital adequacy.

Booth et al (1999: 98-103) trace the development of asset and liability modelling, which has

now become a standard method of optimising a portfolio in the presence of liabilities. The

optimisation problem was first set out by Wise (1984) and was further developed by Wilkie

(1985) and Sherris (1992).

In broad outline, the actuarial contribution to portfolio optimisation followed the portfolio

theory of Markowitz (op. cit.), but allowed explicitly for the investor’s liabilities. Initially, in

Wise and Wilkie (op. cit.), an efficient frontier was developed in mean-variance space defined

as the mean and variance of the surplus at a suitable time horizon. Panjer (1998: 397-405)

gives a modification of this approach, also in mean-variance space. Sherris (op. cit.)

subsequently generalised the problem beyond the constraints of mean-variance analysis.

This also permitted the use of non-quadratic utility functions and non-elliptical distributions

of returns on investments and growth of liabilities. Because the time horizons of investors

other than banks, general insurers and unit trusts typically extend over many years,

multivariate stochastic models of investment returns, and of variables (such as inflation rates)

driving investors’ liabilities, were developed. These include Wilkie’s (1986, 1995) and Smith’s

(1996) model, the TY model (Yakoubov, Teeger & Duval, 1999) and the Whitten & Thomas

35

(2000) model, all for the United Kingdom. Models developed for other countries include

Carter (1991) for Australia, Sharp (1994a, 1994b) and Deaves (1994) for Canada and

Thomson (1996) for South Africa. There is room for improvement in all the published models

to incorporate recent developments in modelling techniques. Numerous unpublished models

have also been developed in-house by commercial organisations.

Because of the complexity of the liabilities of long-term investors and of the stochastic

models, it is not generally possible to obtain solutions to the optimisation problem in closed

form. However, the models facilitate the generation of pseudo-random samples of the surplus

at an appropriate time horizon, which may be used to select portfolios (and other decision

variables) to maximise the expected utility of that surplus. In practice, the utility functions of

the investors (who may, for example, be a board of trustees1) is indeterminate. Under these

circumstances it may be necessary to produce an efficient frontier of the surplus at the time

horizon in mean-variance space (ignoring the effects of higher moments) and to present

alternative portfolios to the trustees along that frontier.

Since it is the trustees who must decide on the investment allocations, it is the trustees who

must define the objective function for the ALM, i.e. the surplus or funding ratio at a time